In a school with 720 students, the ratio of the number of boys to the number of girls is 7 : 5. How many additional girls must be admitted so that the ratio of boys to girls becomes 1 : 1?

Introduction / Context:
This ratio and proportion question is about adjusting the composition of a group to reach a desired ratio. The school initially has a certain ratio of boys to girls, and we are asked how many more girls must be added to make the numbers of boys and girls equal, which corresponds to a ratio of 1 : 1.

Given Data / Assumptions:

• Total number of students in the school is 720.• The ratio of boys to girls initially is 7 : 5.• Only girls are added; the number of boys remains the same.• After adding some girls, the ratio of boys to girls must become 1 : 1.

Concept / Approach:
When a ratio a : b represents two parts of a whole, the actual numbers can be written as 7k and 5k for some integer k. Here, the total number of students is 720, which equals the sum of the numbers of boys and girls. So we first solve for k. Once we know the exact counts of boys and girls, we can determine how many additional girls need to be added so that the number of girls equals the number of boys. This illustrates how ratios translate into real counts and how to adjust one component to reach a target ratio.

Step-by-Step Solution:
Step 1: Let the number of boys be 7k and the number of girls be 5k.Step 2: Total students are 7k + 5k = 12k.Step 3: We are told that 12k = 720.Step 4: Solve for k: k = 720 / 12 = 60.Step 5: Number of boys = 7k = 7 * 60 = 420.Step 6: Number of girls initially = 5k = 5 * 60 = 300.Step 7: We want the new number of girls to be equal to the number of boys, which is 420.Step 8: Therefore, additional girls needed = desired girls − current girls = 420 − 300 = 120.

Verification / Alternative check:
After adding 120 girls, the number of girls becomes 300 + 120 = 420. The number of boys remains 420. Thus, boys : girls becomes 420 : 420, which simplifies to 1 : 1. This confirms that adding 120 girls achieves the target ratio. There is no contradiction with the total because the school size would then become 720 + 120 = 840, which is acceptable since the question allows for additional admissions.

Why Other Options Are Wrong:
• 90: Adding 90 girls would give 390 girls, while boys remain 420, leading to a ratio of 420 : 390, which does not simplify to 1 : 1.• 220: This would result in 520 girls, exceeding the 420 boys and giving a ratio different from 1 : 1.• 240: This would produce 540 girls and 420 boys, again not leading to equality of numbers.

Common Pitfalls:
Some students mistakenly assume the total number of students must remain 720 and try to adjust both boys and girls, which is not what the problem states. Others might miscalculate k by incorrectly dividing 720 by something other than the sum of the ratio parts. It is also easy to confuse the required final ratio and think that the number of boys should change, when in fact only girls are being admitted. Carefully reading the question and systematically using the ratio representation avoids these errors.

Final Answer:
The number of additional girls that must be admitted is 120.